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 \def\ANSWER[#1]{\uncover<2->{{\color{dblue} #1}}}
\title{Combinatorics of affine Hecke algebras of type C.}
\author[Zajj Daugherty]{Zajj Daugherty\\
(joint with Arun Ram)}
\date{May 15, 2013}
\begin{document}

\begin{frame}
\maketitle
\end{frame}
\begin{frame}{}
The \alert{two-boundary braid group} is the group $\cB_k$ generated by 
$ T_0,  T_1,\ldots,  T_k$,
with relations \begin{equation*}\label{Bdefrels}
\begin{tikzpicture}[line width=1pt]
	\foreach \x in {0, 1, 2, 4,5,6}{
		\draw (\x,0) circle (2.5pt);
		\node(\x) at (\x,0){};
		}
	\foreach \x in {0, 1, 2}
		{\node[label=above:$ T_\x$] at (\x){};}
	\node[label=above:$ T_{k-2}$] at (4){};
	\node[label=above:$ T_{k-1}$] at (5){};
	\node[label=above:$ T_{k}$] at (6){};
	\draw[double distance = 2pt] (0)--(1) (5)--(6);
	\draw (1)--(2) (4)--(5);
	\draw[dashed] (2)--(4);
	\only<3>{
		\draw[structure] (1,0) circle (2.5pt);
		\draw[structure] (2,0) circle (2.5pt);
		\draw[structure] (1)--(2);
	}
	\only<4>{
		\draw[structure] (1,0) circle (2.5pt);
		\draw[structure] (0,0) circle (2.5pt);
		\draw[structure, double distance = 2pt] (1)--(0);
	}
\end{tikzpicture}\end{equation*}


\uncover<2->{
Pictorially, \only<1-2>{the generators of $\cB_k$ are identified with the diagrams
$$
{\def\TOP{2}\def\K{6}
\begin{tikzpicture}[scale=.5, line width=1pt]
\node at (0,.5*\TOP) {$ T_k=$};
%%%%%%%%%%%%%
\pgftransformxshift{1.2cm}
%%%%%%%%%%%%%
\SymPolesTop[\TOP][\K]
\SymPolesBottom[0][\K]
\RPole[0,2]
\Under[\K,0][\K+1,1]
\RRPoleO[0,1][\K]
\RRPole[1,2][\K]
\Over[\K+1,1][\K,2]
 \foreach \x in {1,...,5} {
	 \draw (\x,0) -- (\x,\TOP);
	 }
\Bottoms[0][\K]
\Tops[\TOP][\K]
%%%%%%%%%%%%%
\pgftransformxshift{\K cm}
\pgftransformxshift{1.25 cm}
%%%%%%%%%%%%%
\node[right]  at (0, .5*\TOP)  {,};
\end{tikzpicture}
	\qquad 
\begin{tikzpicture}[scale=.5, line width=1pt]]
\node at (0,.5*\TOP) {$ T_0=$};
%%%%%%%%%%%%%
\pgftransformxshift{1.2cm}
%%%%%%%%%%%%%
	\SymPolesTop[\TOP][\K]
	\SymPolesBottom[0][\K]
	\RRPole[0,2][\K]
	\RPole[0,1]
	\Over[1,0][0,1]
	\Under[0,1][1,2]
	\RPoleO[1,2]
	 \foreach \x in {2,...,\K} {
	 \draw (\x,0) -- (\x,\TOP);
	 }
	\Bottoms[0][\K]
	\Tops[\TOP][\K]
%%%%%%%%%%%%%
\pgftransformxshift{\K cm}
\pgftransformxshift{1.25 cm}
%%%%%%%%%%%%%
\node[right]  at (0, .5*\TOP)  {,};
\end{tikzpicture}
}$$
and
$$
{\def\TOP{2} \def\K{6}
\begin{tikzpicture}[scale=.5, line width=1pt]]
\node at (0,.5*\TOP) {$ T_i=$};
%%%%%%%%%%%%%
\pgftransformxshift{1.2cm}
%%%%%%%%%%%%%
	\SymPolesTop[\TOP][\K]
	\SymPolesBottom[0][\K]
	\RRPole[0,2][\K]
	\RPole[0,2]
	\Under[3,0][4,2]
	\Over[4,0][3,2]
	 \foreach \x in {1,2,5,\K} {
		 \draw (\x,0) -- (\x,\TOP);
		 }
	\Bottoms[0][\K]
	\Tops[\TOP][\K]
	\Label[0-.2,\TOP+.25][3][\footnotesize $i$]
	\Label[0-.2,\TOP+.2][4][\footnotesize$i$+1]
%%%%%%%%%%%%%
\pgftransformxshift{\K cm}
\pgftransformxshift{1.25 cm}
%%%%%%%%%%%%%
\node[right]  at (0, .5*\TOP)  {\qquad for $i=1, \dots, k-1$.};
\end{tikzpicture}
}
$$}}
\only<3->{
\uncover<3->{
$$
{\def\TOP{2.5}\begin{tikzpicture}[line width=1pt , scale=.7]
	\node[left] at (.5,.75) {$T_i T_{i+1} T_i ~=$};
	\Under[2,1.5][1,.75]
	\Under[3,1.2][1,0]
	\Over[1,1.5][3,.3]
	\Over[1,.75][2,0]
	\draw(3,1.5)--(3,1.2) (3,.3)--(3,0);
	\Nodes[0][3]
	\Nodes[1.5][3]
\pgftransformxshift{3cm};
\node at (1,.75) {$=$};
\pgftransformxshift{1cm};
	\Under[3,.75][2,0]
	\Under[3,1.5][1,.3]
	\Over[1,1.2][3,0]
	\Over[2,1.5][3,.75]
	\draw(1,1.5)--(1,1.2) (1,.3)--(1,0);
	\Nodes[0][3]
	\Nodes[1.5][3]
\pgftransformxshift{3.5cm};
	\node[right] at (0,.75) {$= ~T_{i+1} T_i T_{i+1}$};
\end{tikzpicture}}$$
}%
\uncover<4->{
$${\def\TOP{2.5}\begin{tikzpicture}[line width=1pt , scale=.7]
\node[left] at (-.5,.5*\TOP) {$T_0 T_1 T_0 T_1~=$};
\PoleTop[\TOP]
	\PoleBottom[0]
	\Under[.2,.5][2,1.5]
	\Under[0,1.5][2,2.5]
	\RPoleO[0,\TOP]
	\Over[0,1.5][2,.7]
	\Over[1,0][.2,.5]
	\draw (2,.7)--(2,0);
	\Over[1,2.5][2,1.7]
	\draw (2,1.7)--(2,1.5);
	\Nodes[\TOP][2]
	\Nodes[0][2]
\pgftransformxshift{3cm};
\node at (0,.5*\TOP) {$=$};
\pgftransformxshift{1cm};
\PoleTop[\TOP]
	\PoleBottom[0]
	\Under[1,2.5][.2,2]
	\Under[2,2][0,1]
	\Under[2,.7][1,0]
	\RPoleO[0,\TOP]
	\Over[.2,2][2,1.2]
	\Over[0,1][2,0]
	\draw (2,2.5)--(2,2);
%	\Over[1,2][2,1.2]
	\draw (2,1.2)--(2,.7);
	\Nodes[\TOP][2]
	\Nodes[0][2]
\pgftransformxshift{2.5cm};
\node[right] at (0,.5*\TOP) {$= ~ T_1 T_0 T_1 T_0$};
\end{tikzpicture}}$$

\centerline{\small (similar picture for $T_k T_{k-1} T_k T_{k-1} = T_{k-1} T_k T_{k-1} T_k$)}
}}

\end{frame}

\begin{frame}{}
The \alert{two-boundary braid group} is the group $\cB_k$ generated by 
$ T_0,  T_1,\ldots,  T_k$,
with relations
\begin{equation*}\label{Bdefrels}
\begin{tikzpicture}[line width=1pt]
	\foreach \x in {0, 1, 2, 4,5,6}{
		\draw (\x,0) circle (2.5pt);
		\node(\x) at (\x,0){};
		}
	\foreach \x in {0, 1, 2}
		{\node[label=above:$ T_\x$] at (\x){};}
	\node[label=above:$ T_{k-2}$] at (4){};
	\node[label=above:$ T_{k-1}$] at (5){};
	\node[label=above:$ T_{k}$] at (6){};
	\draw[double distance = 2pt] (0)--(1) (5)--(6);
	\draw (1)--(2) (4)--(5);
	\draw[dashed] (2)--(4);
\end{tikzpicture}\end{equation*}

\alert{\bf Two (isomorphic) quotients, two perspectives:}\\

\vfill\pause
\alert{1. } Fix $ t, t_0, t_k \in \CC^\times$. The \alert{affine Hecke algebras of type C} $H_k$ is the quotient of $\CC\cB_k$ by
$$\hspace{-.25in}\alert{(*)} \quad 
0=( T_0 - t_0) ( T_0 - t_0^{-1}) = ( T_k - t_k) ( T_k - t_k^{-1}) = ( T_i - t^{1/2}) ( T_i + t^{-1/2})$$
for $i= 1, \dots, k-1$.
\pause


\alert{2. } Let $A, B, C$ be finite dim'l $U_q \fg$-modules. Then $\CC\cB_k$ acts on 
$$B \otimes \underbrace{C \otimes \cdots \otimes C}_{k \text{ factors}} \otimes A$$
Under good (to be defined) conditions, this action factors through the quotient $\alert{(*)}$.\\\pause

\alert{\bf Goal today:} \\
Tell you 3 descriptions of  ``calibrated'' irreducible reps of $H_k$.
\end{frame}




\begin{frame}
The Hecke algebra $H_k$ features invertible, pairwise commuting elements \alert{$Y_1, \dots, Y_k$} (weight lattice part),  \pause and $\CC[ Y_1^{\pm 1}, \dots, Y_k^{\pm 1}]$ carries an action by the \alert{Weyl group $W$ of type C}:\\\pause
\smallskip

$W$ is a group of \alert{signed permutations} generated by transpositions $s_0, s_1, \dots, s_{k-1}$
with relations 
$$\begin{matrix}
\begin{tikzpicture}[scale=.75, line width=1pt]
	\foreach \x in {0, 1, 2, 4,5}{
		\draw (\x,0) circle (2.5pt);
		\node(\x) at (\x,0){};
		}
	\foreach \x in {0, 1, 2}
		{\node[label=above:{\small $s_\x$}] at (\x){};}
	\node[label=above:{\small $s_{k-2}$}] at (4){};
	\node[label=above:{\small $s_{k-1}$}] at (5){};
	\draw[double distance = 2pt] (0)--(1);
	\draw (1)--(2) (4)--(5);
	\draw[dashed] (2)--(4);
\end{tikzpicture}\\
s_j^2=1, \qquad s_0s_1s_0s_1 = s_1s_0s_1s_0 \qquad s_i s_{i+1} s_i = s_{i+1}s_is_{i+1}\end{matrix}$$
for all $j$ and for $i\neq 0$. \pause

The group $W$ acts on $\{-k, \dots, -1, 1, \dots , k\}$ by \\
$\qquad\qquad${$s_0$ swaps 
$
\begin{matrix}\begin{tikzpicture}[scale = .75]
\node[left] (1a) at (0,.75) {\footnotesize$1$};
\node[right] (1b) at (1,.75) {\footnotesize$1$};
\node[left] (n1a) at (0,0) {\footnotesize-$1$};
\node[right] (n1b) at (1,0) {\footnotesize-$1$};
\draw[->] (1a) -- (n1b);
\draw[->] (n1a) -- (1b);
\end{tikzpicture}
\end{matrix}
$
and fixes $i \neq 0$ and }\\
$\qquad\qquad${$s_i$ swaps
$
\begin{matrix}\begin{tikzpicture}[scale = .75]
\node[left] (1a) at (0,.75) {\footnotesize$i$};
\node[right] (1b) at (1,.75) {\footnotesize$i$};
\node[left] (n1a) at (0,0) {\footnotesize$i+1$};
\node[right] (n1b) at (1,0) {\footnotesize$i+1$};
\draw[->] (1a) -- (n1b);
\draw[->] (n1a) -- (1b);
\end{tikzpicture}
\end{matrix}
$
and $
\begin{matrix}\begin{tikzpicture}[scale = .75]
\node[left] (1a) at (0,.75) {\footnotesize$-i$};
\node[right] (1b) at (1,.75) {\footnotesize$-i$};
\node[left] (n1a) at (0,0) {\footnotesize$-i-1$};
\node[right] (n1b) at (1,0) {\footnotesize$-i-1$};
\draw[->] (1a) -- (n1b);
\draw[->] (n1a) -- (1b);
\end{tikzpicture}
\end{matrix}
$
}\\\pause

$W$ acts on the subscripts of the $Y_i$'s with \alert{$Y_{-i} = Y_i^{-1}$}. 
\end{frame}


\begin{frame}{\bf Central characters}



The center of $H_k$ is symmetric Laurent polynomials
$$Z(H_k) = \CC[ Y_1^{\pm 1}, \dots, Y_k^{\pm 1}]^{W}$$ 
 \pause
We can encode central characters as maps
$$\gamma: \{Y_1^{\pm 1}, \dots, Y_k^{\pm 1}\} \to \CC$$
with equivalence under $W$ action; \pause
i.e. $k$-tuples\\
\medskip
\centerline{$\GR{4-}{\gamma = (\gamma_1, \dots, \gamma_{k}) \quad \text{ with } \quad \gamma(Y_i^{\pm 1}) = (\gamma_{i})^{\pm 1}}$}
\pause
{\medskip
\centerline{$\cc = (c_1, \dots, c_{k}) \quad \text{ with } \quad \gamma(Y_i^{\pm 1}) = t^{\pm c_i}$}}
\medskip
(when $\cc$ is real, favorite representatives satisfy $0\leq c_1 \leq \cdots \leq c_k.$)
\bigskip

\pause

\alert{Calibrated} means the $Y_i$'s are all diagonalized.

\bigskip

\pause
\alert{Description 1:} 
Central characters are indexed by points in $k$ dimensions.  \\
\end{frame}

\begin{frame}{Central characters as points \hfill \qquad {\color{black} \small Restrict to real points.}}
{\color{black} \small Fav equivalence class reps:  $0\leq c_1 \leq \cdots \leq c_k.$\\
When $k=2$:}

\centerline{
\begin{tikzpicture}
\uncover<1-2, 7>{\filldraw[gray!30] (0,3)--(0,0)--(3,3)--(0,3);}
\uncover<1>{\draw[thick, dotted] (-3,-3)--(3,3) node[above right] {\tiny $c_1 = c_2$};}
\uncover<1-2>{\filldraw[dgreen] (1,2) circle (2pt) node[above]{\tiny$(c_1, c_2)$};}
\uncover<1>{\draw[thick, ->] (0,-3)--(0,3);
			\draw[thick, ->] (-3,0)--(3,0);}
\uncover<2->{
\draw[thick] (0,-3)--(0,3) node[above] {$\fh^{\alpha_2}$};
\draw[thick] (-3,0)--(3,0) node[right] {$\fh^{\alpha_2+2\alpha_1}$};
\draw[thick] (3,-3)--(-3,3) node[above] {$\fh^{\alpha_1+\alpha_2}$};
\draw[thick] (-3,-3)--(3,3) node[above right] {$\fh^{\alpha_1}$};}
\uncover<3-4>{
	\draw[dotted, thick] (3-.4,3)--(-3,-3+.4) node[left]{\tiny$c_2=c_1+1$};}
\uncover<4>{
	\draw[dotted, thick] (-3+.4,3)--(3,-3+.4) node[right]{\tiny$c_2=-c_1+1$};
	\draw[dotted, thick] (-3,3-.4)--(3-.4,-3) node[below]{\tiny$c_2=-c_1+1$};
	\draw[dotted, thick] (3,3-.4)--(-3+.4,-3) node[below]{\tiny$c_2=c_1-1$};}
\uncover<5->{
	\draw[dotted, thick, dgrey] (3-.4,3)--(-3,-3+.4) node[left]{\tiny$c_2=c_1+1$};
	\draw[dotted, thick, dgrey] (-3+.4,3)--(3,-3+.4) node[right]{\tiny$c_2=-c_1+1$};
	\draw[dotted, thick, dgrey] (-3,3-.4)--(3-.4,-3) node[below right]{\tiny$c_2=-c_1-1$};
	\draw[dotted, thick, dgrey] (3,3-.4)--(-3+.4,-3) node[below left]{\tiny$c_2=c_1-1$};
	\draw[dashed] (3,.8)--(-3,.8) node[left] {\tiny$c_2=r_1$};
	\draw[dashed] (3,2.2)--(-3,2.2) node[left] {\tiny$c_2=r_2$};
	\draw[dashed] (.8,-3)--(.8,3) node[above] {\tiny$c_1=r_1$};
	\draw[dashed] (2.2,-3)--(2.2,3) node[above] {\tiny$c_1=r_2$};
	}
\uncover<6->{
	\draw[dashed] (3,-.8)--(-3,-.8) node[left] {\tiny$c_2=-r_1$};
	\draw[dashed] (3,-2.2)--(-3,-2.2) node[left] {\tiny$c_2=-r_2$};
	\draw[dashed] (-.8,3)--(-.8,-3.2) node[below] {\tiny$c_1=-r_1$};
	\draw[dashed] (-2.2,3)--(-2.2,-3.2) node[below] {\tiny$c_1=-r_2$};
}
\end{tikzpicture}}
{\vspace{-.09in} \scriptsize  \uncover<5->{The $r_i$s depend on $H_k$'s parameters $t_0$ and $t_k$: $r_1 = \log_t(t_0/t_k)$, $r_2 = \log_t(t_0t_k)$}}
\end{frame}


\begin{frame}{Central characters as points; \\
Calibrated reps as ``skew local regions''}

\centerline{
\begin{tikzpicture}[scale=1.875]
	\filldraw[gray!20] (0,3)--(0,0)--(3,3)--(0,3);
\draw[thick] (0,-.2)--(0,3) node[above] {$\fh^{\alpha_2}$};
\draw[thick] (-.2,-.2)--(3,3) node[above right] {$\fh^{\alpha_1}$};
\draw[gray] (-.2,0)--(3,0) node[right] {\color{gray}$\fh^{\alpha_2+2\alpha_1}$};
	\draw[dotted, thick, black!70] (3-.4,3)--(-.2,-.2+.4) node[left]{\tiny$c_2=c_1+1$};
	\draw[dotted, thick, black!70] (-.2,.2+.4)--(.4+.2,-.2) node[right]{\tiny$c_2=-c_1+1$};
	\draw[dotted, thick, black!70] (3,.6)--(-.2,.6) node[left] {\tiny$c_2=r_1$};
	\draw[dotted, thick, black!70] (3,2.4)--(-.2,2.4) node[left] {\tiny$c_2=r_2$};
	\draw[dotted, thick, black!70] (.6,-.2)--(.6,3) node[above] {\tiny$c_1=r_1$};
	\draw[dotted, thick, black!70] (2.4,-.2)--(2.4,3) node[above] {\tiny$c_1=r_2$};
\uncover<4->{
	\filldraw[red!50] (1.3+.2,2.4)--(1.3+.15,2.4+.15)--(1.3,2.4+.2)--(1.3-.15,2.4+.15)--(1.3-.2,2.4);
			\filldraw[yellow!70] (1.3+.2,2.4)--(1.3+.15,2.4-.15)
							--(1.3,2.4-.2)--(1.3-.15,2.4-.15)--(1.3-.2,2.4);}
			\uncover<3->{\REG[1.3,2.4]}
\uncover<8->{	\filldraw[blue!50] (1+.2,2)--(1+.15,2+.15)--(1,2+.2)
					--(1-.15,2+.15)--(1-.2,2)--(1-.15,2-.15)
					--(1,2-.2)--(1+.15,2-.15); 
					\REG[1,2]}
\uncover<6->{	\filldraw[blue!50] (.6+.15,1+.15)--(.6,1+.2)--(.6,1); 
			\filldraw[green!50] (.6,1+.2)--(.6-.15,1+.15)--(.6-.2,1)--(.6-.15,1-.15)--(.6,1);
			\filldraw[yellow!70] (.6-.15,1-.15)--(.6,1-.2)--(.6,1);
			\filldraw[red!50](.6,1-.2)--(.6+.15,1-.15)--(.6+.2,1)--(.6+.15,1+.15)--(.6,1); }
			\uncover<5->{\REG[.6,1]}
\uncover<7->{	\filldraw[red!50] (.6,2.4)--(.6,2.4+.2)--(.6-.15,2.4+.15)--(.6-.2,2.4);
		\filldraw[yellow!70] (.6,2.4)--(.6-.2,2.4)--(.6-.15,2.4-.15)--(.6,2.4-.2);
		\filldraw[green!50] (.6,2.4)--(.6,2.4-.2)--(.6+.15,2.4-.15)--(.6+.2,2.4);
		\filldraw[blue!50] (.6,2.4)--(.6+.2,2.4)--(.6+.15,2.4+.15)--(.6,2.4+.2);
		\REG[.6,2.4]}
\uncover<9->{%	\filldraw[red!50] (2.4,2.4)--(2.4-.2,2.4)--(2.4-.15,2.4+.15)--(2.4,2.4+.2)--(2.4,2.4);
		\REGnw[2.4,2.4]
		\REGe[0,1.7]}
\uncover<2->{
	% 2
	\filldraw[structure] (.6,2.4) circle (1pt) node[above]{};
	% 3/1
	\filldraw[structure] (.2,.6) circle (1pt) node[above]{};
	\filldraw[structure] (.6,1) circle (1pt) node[above]{};
	\filldraw[structure] (2,2.4) circle (1pt) node[above]{};
	\filldraw[structure] (2.4,2.8) circle (1pt) node[above]{};
	%4:
	\filldraw[structure] (.6,1.7) circle (1pt) node[above]{};
	\filldraw[structure] (1.3,1.7) circle (1pt) node[above]{};
	\filldraw[structure] (1.3,2.4) circle (1pt) node[above]{};
	%8
	\filldraw[structure] (1,2) circle (1pt) node[above]{};
	%Walls
	\filldraw[orange] (0,1.7) circle (1pt) node[above]{};
	\filldraw[orange] (2.4,2.4) circle (1pt) node[above]{};
	}
\end{tikzpicture}}
{ \scriptsize The $r_i$s depend on $H_k$'s parameters $t_0$ and $t_k$: $r_1 = \log_t(t_0/t_k)$, $r_2 = \log_t(t_0t_k)$}
\end{frame}

\begin{frame}{}
\alert{Description 1:} 
Central characters are indexed by points.  \\
Irreps  are indexed by skew local regions around points.\\ 
Basis is indexed by chambers in each region.\\

\pause
\alert{Description 2:} Box arrangements. \\\pause
Start with diagonal lines labeled by $\ZZ$.
\uncover<4->{Restrict to points in $(\ZZ + \beta)^k$. 
A central character $\cc$ gives a list of diagonal placements. }\\
\uncover<5->{For example: \centerline{$\cc = (2,3,4,4,5)$}}


\def\ExA{6}
\def\ExB{7}
\def\ExC{8}
\def\ExD{9}
\def\ExE{10}
\def\ExR{11}
\def\ExRb{12}
\def\FillA{13}
\def\FillB{14}
%\uncover<\ExR->{\centerline{Suppose $r_2 = 5$}}
\begin{tikzpicture}[scale=.5]
\uncover<1-\ExE>{\draw (5,0) node[above]{\tiny 5} to (5+4,-4);}
\uncover<\ExR->{\draw[red!50, thick] (5,0) node[above, draw, circle, inner sep=1pt]{\tiny $r_2$} to (5+4,-4);}
\foreach \x in {1,2,3,4,6}{
\draw (\x,0) node[above]{\tiny \x} to (\x+4,-4);}
\uncover<\ExB>{\BeadB{2}{3.4}	\BeadB{3}{2}	\BeadB{4}{.6}	\BeadB{4}{2.75}	\BeadB{5}{3}}
\uncover<\ExA-\ExB>{\Bead{2}{3.4}{1}	\Bead{3}{2}{2}	\Bead{4}{.6}{3}	\Bead{4}{2.75}{4}	\Bead{5}{3}{5}}
\uncover<\ExC->{
	\BeadB{2}{3}	\BeadB{3}{2}	\BeadB{4}{1}	\BeadB{4}{2}	\BeadB{5}{2}
	\Bead{2}{3}{1}	\Bead{3}{2}{2}	\Bead{4}{1}{3}	\Bead{4}{2}{4}	\Bead{5}{2}{5}
	}
\uncover<\ExRb->{\filldraw[structure, fill=red] (5+2.5,-2.5) circle (5pt);}
\uncover<\ExD->{\node[below, structure] at (5.5,-4){ \footnotesize $1>2,~ 2>3,~ 2<4, ~3,4<5 $};
\draw[very thick, ->, structure] (8,-1) to node[midway, above right, inner sep=1pt]{\tiny $+$} (9,-2);}
\uncover<\ExRb->{\node[below, structure] at (5.5,-5){ \footnotesize   $5<$ \Marker};}
\end{tikzpicture}
\qquad 
\begin{tikzpicture}[scale=.5]
\uncover<\ExE-\FillA>{\uncover<\ExE>{\draw (5,0) node[above]{\tiny 5} to (5+4,-4);}
\uncover<\ExR->{\draw[red!50, thick] (5,0) node[above, draw, circle, inner sep=1pt]{\tiny $r_2$} to (5+4,-4);}
\foreach \x in {1,2,3,4,6}{
\draw (\x,0) node[above]{\tiny \x} to (\x+4,-4);}
\BeadB{2}{2}	\BeadB{3}{2}	\BeadB{4}{1}	\BeadB{4}{2}	\BeadB{5}{1}
\Bead{2}{2}{1}	\Bead{3}{2}{2}	\Bead{4}{1}{3}	\Bead{4}{2}{4}	\Bead{5}{1}{5}
\uncover<\ExRb->{\filldraw[structure, fill=red]  (5+.5,-.5) circle (4pt);}
\node[below, structure] at (5.5,-4){ \footnotesize   $1<2, ~ 2>3, ~ 3<5,~ 4>5 $};
\draw[very thick, ->, structure] (8,-1) to node[midway, above right, inner sep=1pt]{\tiny $+$} (9,-2);
\uncover<\ExRb->{\node[below, structure] at (5.5,-5){ \footnotesize   $5>$ \Marker };}}
\uncover<\FillB->{
\pgftransformyshift{.5cm}
\draw (0,0)--(0,-3)--(1,-3)--(1,0)--(0,0) (0,-1)--(3,-1)--(3,-2)--(0,-2) (2,-1)--(2,-2); \node at (3, -2) {\Marker};
\node at (.5,-.5) {-5}; \node at (.5,-1.5) {-4}; \node at (.5,-2.5) {-3}; \node at (1.5,-1.5) {-2}; \node at (2.5,-1.5) {-1}; 
\node[right] at (3,-1.5){\small ok!};
\pgftransformyshift{-3.5cm}
\draw (0,0)--(0,-3)--(1,-3)--(1,0)--(0,0) (0,-1)--(3,-1)--(3,-2)--(0,-2) (2,-1)--(2,-2); \node at (3, -2) {\Marker};
\node at (.5,-.5) {-5}; \node at (.5,-1.5) {-4}; \node at (.5,-2.5) {1}; \node at (1.5,-1.5) {-3}; \node at (2.5,-1.5) {-2};
\node[right] at (3,-1.5){\small ok!}; 
\pgftransformyshift{3.5cm}
\pgftransformxshift{5.5cm}
\begin{scope}[black!40!white, every node/.style={black!40!white}]
	\draw (0,0)--(0,-3)--(1,-3)--(1,0)--(0,0) (0,-1)--(3,-1)--(3,-2)--(0,-2) (2,-1)--(2,-2); 
	\node at (.5,-.5) {-5}; \node at (.5,-1.5) {-4}; \node at (.5,-2.5) {1}; \node at (1.5,-1.5) {-3}; \node at (2.5,-1.5) {2}; 
	\node[right] at (3,-1.5){\small bad!};
\end{scope}
	\node at (3, -2) {\Marker};
\pgftransformyshift{-3.5cm}
\begin{scope}[black!40!white, every node/.style={black!40!white}]
	\draw (0,0)--(0,-3)--(1,-3)--(1,0)--(0,0) (0,-1)--(3,-1)--(3,-2)--(0,-2) (2,-1)--(2,-2); 
	\node at (.5,-.5) {-5}; \node at (.5,-1.5) {-4}; \node at (.5,-2.5) {5}; \node at (1.5,-1.5) {-3}; \node at (2.5,-1.5) {-2}; 
	\node[right] at (3,-1.5) {\small bad!};
\end{scope}
\node at (3, -2) {\Marker};
}

\end{tikzpicture}

\uncover<\FillA->{Basis indexed by standard fillings with $\{\pm 1, \dots, \pm k\}$ with restrictions: \\
{\small $\quad$ (1) Exactly one of $i$ or $-i$ appears.\\
$\quad$ (2) If $\mathrm{box_i} <$ \Marker \ , then filling is negative. If $\mathrm{box_i} >$ \Marker \ , filling is positive.}}
\end{frame}

\begin{frame}{Points versus box arrangements}

\def\EightA{1}
\def\EightB{2}
\def\FourA{4}
\def\FourB{3}
\def\OneA{5}
\def\OneB{8}

\centerline{
\begin{tikzpicture}[scale=1.875]
	\filldraw[gray!20] (0,3)--(0,0)--(3,3)--(0,3);
\draw[thick] (0,-.2)--(0,3) node[above] {$\fh^{\alpha_2}$};
\draw[thick] (-.2,-.2)--(3,3) node[above right] {$\fh^{\alpha_1}$};
	\draw[dotted, thick, black!70] (3-.4,3)--(-.2,-.2+.4) node[left]{\tiny$c_2=c_1+1$};
	\draw[dotted, thick, black!70] (3,.6)--(-.2,.6) node[left] {\tiny$c_2=r_1$};
	\draw[dotted, thick, black!70] (3,2.4)--(-.2,2.4) node[left] {\tiny$c_2=r_2$};
	\draw[dotted, thick, black!70] (.6,-.2)--(.6,3) node[above] {\tiny$c_1=r_1$};
	\draw[dotted, thick, black!70] (2.4,-.2)--(2.4,3) node[above] {\tiny$c_1=r_2$};
\uncover<\EightA-\EightB>{\filldraw[green!50] (1+.2,2)--(1+.15,2+.15)--(1,2+.2)
					--(1-.15,2+.15)--(1-.2,2)--(1-.15,2-.15)
					--(1,2-.2)--(1+.15,2-.15); 
					\REG[1,2]
					\filldraw[structure] (1,2) circle (1pt) node[above]{};}
\uncover<\FourA>{
	\filldraw[green!50] (1.3+.15,1.7+.15)--(1.3,1.7+.2)--(1.3-.15,1.7+.15)--(1.3-.2,1.7)--(1.3-.15,1.7-.15);
	{\filldraw[yellow!70] (1.3+.15,1.7+.15)--(1.3+.2,1.7)--(1.3+.15,1.7-.15)--(1.3,1.7-.2)--(1.3-.15,1.7-.15);}
	\REG[1.3,1.7]
	\filldraw[structure] (1.3,1.7) circle (1pt) node[above]{};
}
\uncover<\FourB>{			
		\filldraw[yellow!70] (1.3+.2,2.4)--(1.3+.15,2.4-.15)
							--(1.3,2.4-.2)--(1.3-.15,2.4-.15)--(1.3-.2,2.4);
		\filldraw[green!50] (1.3+.2,2.4)--(1.3+.15,2.4+.15)
							--(1.3,2.4+.2)--(1.3-.15,2.4+.15)--(1.3-.2,2.4);
		\REG[1.3,2.4]
		\filldraw[structure] (1.3,2.4) circle (1pt) node[above]{};
}
\uncover<\OneA>{
		\filldraw[blue!50] (2,2.4)--(2-.15,2.4-.15)--(2-.2,2.4);
		\filldraw[orange!70] (2,2.4)--(2+.15,2.4+.15)--(2+.2,2.4);
			\filldraw[green!50] (2-.2,2.4)--(2-.15,2.4+.15)--(2,2.4+.2)--(2+.15,2.4+.15)--(2,2.4);
			\filldraw[yellow!70](2+.2,2.4)--(2+.15,2.4-.15)--(2,2.4-.2)--(2-.15,2.4-.15)--(2,2.4); 
		\REG[2,2.4]
	\filldraw[structure] (2,2.4) circle (1pt) node[above]{};
}
\node at (0,-.5) {};
\end{tikzpicture}
\begin{tikzpicture}[scale=.35]
\uncover<\EightA->{\draw[double] (-3,4)--(-3,-18);}
\uncover<\EightA-\EightB>{
	\draw(-.5,1.5) node[above left, inner sep=1pt]{\tiny$c_1$} --(1.5,-.5);
	\draw(1.5,3.5) node[above left, inner sep=1pt]{\tiny$c_2$} --(3.5,1.5);
		\cBOX{0}{0}{green!50}
		\cBOX{2}{2}{green!50}}
\uncover<\EightB>{
	\foreach \y in {0, 1, ..., 4}{\draw[gray](-2.5,-1.75-4*\y)--(5.25,-1.75-4*\y);}
	\foreach \x in {0,1,2} {\draw[gray](-2.5+3.875*\x,-1.75)--(-2.5+3.875*\x,-17.75);}
	\ShiftX{-2cm} \ShiftY{-5cm}
					\BOX{0}{0}{1}\BOX{1.5}{1.5}{2}
	 \ShiftY{-4cm}		\BOX{0}{0}{2}\BOX{1.5}{1.5}{1}
	 \ShiftY{-4cm}		\BOX{0}{0}{-1}\BOX{1.5}{1.5}{-2}
	 \ShiftY{-4cm}		\BOX{0}{0}{-2}\BOX{1.5}{1.5}{-1}
	\ShiftX{4cm} \ShiftY{12cm}
					\BOX{0}{0}{-1}\BOX{1.5}{1.5}{2}
	 \ShiftY{-4cm}		\BOX{0}{0}{2}\BOX{1.5}{1.5}{-1}
	 \ShiftY{-4cm}		\BOX{0}{0}{1}\BOX{1.5}{1.5}{-2}
	 \ShiftY{-4cm}		\BOX{0}{0}{-2}\BOX{1.5}{1.5}{1}
	 \ShiftX{-2cm} \ShiftY{14cm}}
\uncover<\FourA>{
	\draw(-.5,1.5) node[above left, inner sep=1pt]{\tiny$c_1$} --(1.5,-.5);
	\draw(.5,1.5) node[above left, inner sep=1pt]{\tiny$c_2$} --(2.5,-.5);
		\cBOX{0}{0}{green!50}\cBOX{1}{0}{green!50}
	\ShiftY{-8cm}	
	\draw(-.5,1.5) node[above left, inner sep=1pt]{\tiny$c_1$} --(1.5,-.5);
	\draw(-.5,2.5) node[above left, inner sep=1pt]{\tiny$c_2$} --(1.5,.5);
		\cBOX{0}{0}{yellow!70}\cBOX{0}{1}{yellow!70}
	\ShiftY{8cm}}
\uncover<\FourB>{
	\draw(-.5,1.5) node[above left, inner sep=1pt]{\tiny$c_1$} --(1.5,-.5);
	\draw(1.5,3.5) node[above left, inner sep=1pt]{\tiny$r_2$} --(3.5,1.5);
		\cBOX{0}{0}{green!50}\cBOX{2}{2}{green!50}
		\filldraw[black, fill=red]  (2,3) circle (6pt);
	\ShiftY{-8cm}	
	\draw(-.5,1.5) node[above left, inner sep=1pt]{\tiny$c_1$} --(1.5,-.5);
	\draw(1.5,3.5) node[above left, inner sep=1pt]{\tiny$r_2$} --(3.5,1.5);
		\cBOX{0}{0}{yellow!70}\cBOX{2}{2}{yellow!70}
		\filldraw[black, fill=red]  (3,2) circle (6pt);	\ShiftY{11cm}}
\uncover<\OneA>{
	\draw(-.5,1.5) node[above left, inner sep=1pt]{\tiny$c_1$} --(1.5,-.5);
	\draw(.5,1.5) node[above left, inner sep=1pt]{\tiny$r_2$} --(2.5,-.5);
		\cBOX{0}{0}{green!50}\cBOX{1}{0}{green!50}
		\filldraw[black, fill=red]  (1,1) circle (6pt);	
	\ShiftY{-5cm}
	\draw(-.5,1.5) node[above left, inner sep=1pt]{\tiny$c_1$} --(1.5,-.5);
	\draw(.5,1.5) node[above left, inner sep=1pt]{\tiny$r_2$} --(2.5,-.5);
		\cBOX{0}{0}{blue!30}\cBOX{1}{0}{blue!30}
		\node[] at (.5,.5) {\small -$2$};  \node[] at (1.5,.5) {\small -$1$};
		\filldraw[black, fill=red]  (2,0) circle (6pt);	
	\ShiftY{-5cm}		
	\draw(-.5,1.5) node[above left, inner sep=1pt]{\tiny$c_1$} --(1.5,-.5);
	\draw(-.5,2.5) node[above left, inner sep=1pt]{\tiny$r_2$} --(1.5,.5);
		\cBOX{0}{0}{yellow!70}\cBOX{0}{1}{yellow!70}
		\filldraw[black, fill=red]  (1,1) circle (6pt);	
	\ShiftY{-5cm}
	\draw(-.5,1.5) node[above left, inner sep=1pt]{\tiny$c_1$} --(1.5,-.5);
	\draw(-.5,2.5) node[above left, inner sep=1pt]{\tiny$r_2$} --(1.5,.5);
		\cBOX{0}{0}{orange!50} \cBOX{0}{1}{orange!50}
		\node[] at (.5,.5) {\small $2$};  \node[] at (.5,1.5) {\small $1$};
		\filldraw[black, fill=red]  (0,2) circle (6pt);	
	\ShiftY{-5cm}}
\end{tikzpicture}
}
\end{frame}


\begin{frame}{}
\alert{Description 1:} 
Central characters are indexed by points.  \\
Irreps  are indexed by skew local regions around points.\\ 
Basis is indexed by chambers in each region.\\
\medskip

\alert{Description 2:} Marked box arrangements. \\
Basis indexed by good fillings.\\\medskip

\pause

\alert{Description 3:} Partitions.\\
Representation arise in Schur-Weyl duality with certain $U_q\fgl_n$ reps. 

\end{frame}



\begin{frame}{Centralizer properties}
Let $U = U_q \fgl_n$ be the quantum group for $\fgl_n(\CC)$. 
We're interested in certain finite dimensional simple $U$-modules $\alert{L(\lambda)}$ indexed by  \alert{partitions:}\\
\centerline{ \begin{tikzpicture}[xscale=.4, yscale=-.4]
	\node[left] at (-.6,1.5){$\lambda =$};
		\Part{4,3,1}
	\uncover<3->{
		\foreach \x in {0,1,2,3} {\draw[structure] (\x, 0) to (\x-.3, -.3) node[inner sep=1pt, above left] {\tiny \x};}
		\foreach \x in {1,2} {\draw[structure] (0, \x) to (-.3, \x-.3) node[inner sep=1pt, above left] {\tiny -\x};}
		}
		\end{tikzpicture}}
		
\centerline{\small (drawn as a collection of boxes piled up and to the left)}

\pause
\vfill

In particular, rectangular partitions:
$$
	(a^c) = \begin{matrix}\begin{tikzpicture}[scale=.4]
		\draw (0,0)--(4,0)--(4,3)--(0,3)--(0,0);
		\foreach \x in {1,2,3}{\draw[black!70] (\x,.2)--(\x,2.8);}
		\foreach \x in {1,2}{\draw[black!70] (.2,\x)--(3.8,\x);}
		\node[left] at (0,1.5){$c$};
		\node[above] at (2,3) {$a$};
		\uncover<4->{
			\draw[thick, dotted, structure] (4,3)--(5,3)--(5,2)--(4,2) (0,0)--(0,-1)--(1,-1)--(1,0);
			\node[structure]  at (4.5, 2.5) {\footnotesize $a$};\node[structure]  at (.5,-.5) {\footnotesize -$c$};
		}					
	     \end{tikzpicture} \end{matrix}
     $$
\pause

%$H_k$ has a commuting action with $U$ on the space \\
%\centerline{$L((a^c)) \otimes L((b^d)) \otimes \big(L(\tikz[scale=.2]{\BOX{0}{0}{}})\big)^{\otimes k}$.}\pause

The \alert{content} of a box is its diagonal number.\pause

The eigenvalues of $T_0$ and $T_k$ are controlled by the contents of addable boxes to $(a^c)$ and $(b^d)$. 



\end{frame}

\begin{frame}{Centralizer properties}
\begin{thm}[\small D.-Ram]
\begin{enumerate}
\item Let $U = U_q \fg$, and let $A$, $B$, and $C$ be finite dim'l $U$-modules.  The two-boundary braid group $\cB_k$ acts on $B \otimes (C)^{\otimes k} \otimes A$
(via $R$-matrices) and this action commutes with that of $U$. \pause

\centerline{\def\K{5}
\def\TOP{1}
$\begin{tikzpicture}[yscale=.8,line width=1pt ]
\SymPolesTop[\TOP][\K]
\SymPolesBottom[0][\K]
\draw[style=over] (1,\TOP)--(3,0);
\Over[3,\TOP][.15, .3*\TOP]
\RPoleO[0,\TOP]
\Over[.15, .3*\TOP][1,0]
\draw[style=over] (4,\TOP)--(2,0); 
\Under[5,\TOP][5.85,.5*\TOP] 
\RRPoleO[0,\TOP][\K]
\Over[5.85,.5*\TOP][5,0]
\draw[style=over](2,\TOP)--(4,0);
\foreach \x in {1,...,5}{
 \filldraw [black] (\x,\TOP) circle (2pt);
 \filldraw [black] (\x,0) circle (2pt);
}
  \node at (.415, \TOP+.5) {\scriptsize $B$};
  \node at (.415, -.5) {\scriptsize $B$};
    \node at (5.565, \TOP+.5) {\scriptsize $A$};
  \node at (5.565, -.5) {\scriptsize $A$};
  \node at (.715, \TOP+.5) {\scriptsize $\otimes$};
  \node at (.715, -.5) {\scriptsize $\otimes$};
  \node at (5.3, \TOP+.5) {\scriptsize $\otimes$};
  \node at (5.3, -.5) {\scriptsize $\otimes$};
  \foreach \x in {1,...,5}{
 	\node at (\x, \TOP+.5) {\scriptsize $C$};
 	\node at (\x, -.5) {\scriptsize $C$};
  }
  \foreach \x in {1,...,4}{
 	\node at (\x+.5, \TOP+.5) {\scriptsize $\otimes$};
 	\node at (\x+.5, -.5) {\scriptsize $\otimes$};
  }
\end{tikzpicture}$}
\end{enumerate}
\end{thm}
\pause
$R$-matrices: $U$ has an associated invertible element $R=\sum_{\cR} R_1\otimes R_2$ of $U \otimes U$ that gives us a map\\
\centerline{\begin{tikzpicture}[xscale=.8, yscale=.6, line width=1pt ]
	\node at (-3,1.4) 
		{$\check R_{MN}\colon M\otimes N \longrightarrow N\otimes M$};
	\Under[1,1][2,2] 	\Over[1,2][2,1]
	\Nodes[1][2]	\Nodes[2][2]
	\node at (1.5,2.5) {$M \otimes N$};
	\node at (1.5,.5) {$N \otimes M$};
\end{tikzpicture}}
This map acts on a component $L(\lambda)$ of $L(\mu) \otimes L(\begin{tikzpicture}[scale=.25]\draw (0,0)--(0,1)--(1,1)--(1,0)--(0,0);\end{tikzpicture})$ by $q^{2c(\lambda/\mu)}$. 

\end{frame}


\begin{frame}{Centralizer properties}
\begin{thm}[\small D.-Ram]
\begin{enumerate}
\item[2.] If $\fg = \fgl_n$, $A= L((a^c))$, $B = L((b^d))$, and $C = L(\begin{tikzpicture}[scale=.25]\draw (0,0)--(0,1)--(1,1)--(1,0)--(0,0);\end{tikzpicture})$, then the action in \alert{1.} factors through the quotient by
\end{enumerate}
$$0=(T_0 - t_0)(T_0-t_0^{-1}) = (T_k - t_k)(T_k-t_k^{-1}) = (T_i - t^{1/2})(T_i+t^{-1/2})$$
where $t=q^2$, $t_0 = t^{\frac{1}{2}(b+d)}$, and  $t_k= t^{\frac{1}{2}(a+c)}$.
\end{thm}
\pause



\centerline{\def\TOP{1}\def\K{1}
\begin{tikzpicture}[line width=1pt , scale=.75]
\node[left] at (.3,.5*\TOP) {$T_0: $};
%%%%%%%%%%%%%%
\pgftransformxshift{0cm}
%%%%%%%%%%%%%
	\PoleTop[\TOP]
	\PoleBottom[0]
	\Under[.2,.5][1,1.2]
	\RPoleO[0,1]
	\Over[1,-.2][.2,.5]
	\Nodes[-.2][\K]
	\Nodes[\TOP+.2][\K]
\Label[0-.3,\TOP+.3][.7][\footnotesize $B\!\otimes\!C$]
%%%%%%%%%%%%%
\pgftransformxshift{3.75 cm}
%%%%%%%%%%%%%
\node[left] at (1,.5*\TOP) {$\cdots \qquad T_k:  \cdots$};
%%%%%%%%%%%%%
%\pgftransformxshift{.5cm}
%%%%%%%%%%%%%
%%%%%%%%%%%%%%
\draw (1+.5,\TOP) -- (1+.5,\TOP+.25) (1+.65,\TOP) -- (1+.65,\TOP+.25) ;
\draw (1+.65,\TOP+.25) arc (0:360:2pt and 1.5pt);
\draw (1+.5,0) -- (1+.5,-.25) (1+.65,0) -- (1+.65,-.25) ;
\draw (1+.65,-.25)  arc (0:-180:2pt and 1.5pt);
\Under[\K,-.2][\K+.8,.5]
\RRPoleO[0,1][\K]
\Over[\K+.8,.5][\K,1.2]
\Nodes[0-.2][\K]
\Nodes[\TOP+.2][\K]
\Label[0-.3,\TOP+.3][1.3][\footnotesize $C\!\otimes\!A$]
\pgftransformxshift{3.5 cm}
%%%%%%%%%%%%%
\node[left] at (1,.5*\TOP) {$T_i:$};
%%%%%%%%%%%%%%
%\pgftransformxshift{1.2cm}
%%%%%%%%%%%%%%
	\Under[1,0][2,\TOP]
	\Over[2,0][1,\TOP]
	\Nodes[0][2]
	\Nodes[\TOP][2]
	\Label[0,\TOP][1.5][\footnotesize $C \otimes C$]
\end{tikzpicture}
}

$${\qquad
	\begin{picture}(60, 18)(-10,12)
		\multiput(10,12)(10,0){2} {\line(0,1){16}} %V
		\multiput(2,20)(0,10){1} {\line(1,0){26}} %H
		%
		\thicklines
		{\alert{
		\put(30,20){\dashbox{1}(10,10){\footnotesize $b$}}
		\put(0,0){\dashbox{1}(10,10){\footnotesize -$d$}}
		}}		
		\multiput(0,10)(30,0){2} {\line(0,1){20}} %V
		\multiput(0,10)(0,20){2} {\line(1,0){30}} %H
		%
		\put(-10,15){$d$}
		\put(15,32){$b$}
		%
     \end{picture} 
\qquad\quad
	\begin{picture}(60, 18)(-10,12)
		\multiput(10,2)(10,0){3} {\line(0,1){26}} %V
		\multiput(2,10)(0,10){2} {\line(1,0){36}} %H
		%
		\thicklines
		{\alert{
		\put(40,20){\dashbox{1}(10,10){\footnotesize $a$}}
		\put(0,-10){\dashbox{1}(10,10){\footnotesize -$c$}}
		}}		
		\multiput(0,0)(40,0){2} {\line(0,1){30}} %V
		\multiput(0,0)(0,30){2} {\line(1,0){40}} %H
		%
		\put(-10,12){$c$}
		\put(17,32){$a$}
		%
     \end{picture} 
     \qquad\quad
	\begin{picture}(60, 18)(-10,12)
		\thicklines
		{\alert{
		\put(10,20){\dashbox{1}(10,10){\footnotesize $1$}}
		\put(0,10){\dashbox{1}(10,10){\footnotesize -$1$}}
		}}		
		\multiput(0,20)(10,0){2} {\line(0,1){10}} %V
		\multiput(0,20)(0,10){2} {\line(1,0){10}} %H
     \end{picture} 
     }
$$


\end{frame}

\begin{frame}{Exploring our new favorite tensor space, $A \otimes B \otimes C^{\otimes k}$} 
Move the right pole to the left:
$$
{\def\TOP{1}\def\K{5}
\begin{tikzpicture}[scale=.8, line width = 1pt]
\SymPolesTop[\TOP][\K]
\SymPolesBottom[0][\K]
\RPole[0,\TOP]
\Under[\K,0][\K+1,.5*\TOP]
\RRPoleO[0,.5*\TOP][\K]
\RRPole[.5*\TOP,\TOP][\K]
\Over[\K+1,.5*\TOP][\K,\TOP]
 \foreach \x in {1,...,4} {
	 \draw (\x,0) -- (\x,\TOP);
	 }
\Nodes[0][\K]
\Nodes[\TOP][\K]
  \node at (.415, 1.6) {\scriptsize $B$};
  \node at (.415, -.5) {\scriptsize $B$};
    \node at (5.65, 1.6) {\scriptsize $A$};
  \node at (5.65, -.5) {\scriptsize $A$};
  \node at (.715, 1.6) {\scriptsize $\otimes$};
  \node at (.715, -.5) {\scriptsize $\otimes$};
  \node at (5.3, 1.6) {\scriptsize $\otimes$};
  \node at (5.3, -.5) {\scriptsize $\otimes$};
  \foreach \x in {1,...,5}{
 	\node at (\x, 1.6) {\scriptsize $C$};
 	\node at (\x, -.5) {\scriptsize $C$};
  }
  \foreach \x in {1,...,4}{
 	\node at (\x+.5, 1.6) {\scriptsize $\otimes$};
 	\node at (\x+.5, -.5) {\scriptsize $\otimes$};
  }
%%%%%%%%%%%%%
\pgftransformxshift{\K cm}
\pgftransformxshift{2.5 cm}
%%%%%%%%%%%%%
\node[right]  at (-1.5, .5*\TOP)  {$=$};
\PolesTop[\TOP]
\PolesBottom[0]
\LPole[0,1]
\Over[\K,0][-.5,.5*\TOP]
\Under[-.5,.5*\TOP][\K,\TOP]
\LPoleO[.5*\TOP,\TOP]
\RPoleO[0,\TOP]
 \foreach \x in {1,...,4}  {
	 \draw[thin, style=over] (\x,0) -- (\x,\TOP);
	 }
\Nodes[0][\K]
\Nodes[\TOP][\K]
  \node at (.415, 1.6) {\scriptsize $B$};
  \node at (.415, -.5) {\scriptsize $B$};
    \node at (-.2, 1.6) {\scriptsize $A$};
  \node at (-.2, -.5) {\scriptsize $A$};
  \node at (.715, 1.6) {\scriptsize $\otimes$};
  \node at (.715, -.5) {\scriptsize $\otimes$};
  \node at (.15, 1.6) {\scriptsize $\otimes$};
  \node at (.15, -.5) {\scriptsize $\otimes$};
  \foreach \x in {1,...,5}{
 	\node at (\x, 1.6) {\scriptsize $C$};
 	\node at (\x, -.5) {\scriptsize $C$};
  }
  \foreach \x in {1,...,4}{
 	\node at (\x+.5, 1.6) {\scriptsize $\otimes$};
 	\node at (\x+.5, -.5) {\scriptsize $\otimes$};
  }
\end{tikzpicture}
}
$$

\uncover<2->{
New favorite generators: 
$${\def\TOP{1}\def\K{1}
\begin{tikzpicture}[line width=1pt , scale=.75]
\node[left] at (0,.5*\TOP) {$T_0=$};
%%%%%%%%%%%%%
\pgftransformxshift{.5cm}
%%%%%%%%%%%%%
	\PolesTop[\TOP]
	\PolesBottom[0]
	\LPoleO[0,1]
	\Under[.2,.5][1,1]
	\RPoleO[0,1]
	\Over[1,0][.2,.5]
	\Nodes[0][\K]
	\Nodes[\TOP][\K]
%%%%%%%%%%%%%
\pgftransformxshift{2.5 cm}
%%%%%%%%%%%%%
\node at (0,.5*\TOP) {$T_i=$};
%%%%%%%%%%%%%%
%\pgftransformxshift{1.2cm}
%%%%%%%%%%%%%%
	\Under[1,0][2,\TOP]
	\Over[2,0][1,\TOP]
	\Nodes[0][2]
	\Nodes[\TOP][2]
	\Label[0,\TOP][1][\footnotesize $i$]
	\Label[0,\TOP][2][\footnotesize$i$+1]
%%%%%%%%%%%%%
\pgftransformxshift{6 cm}
\pgftransformxshift{.25 cm}
%%%%%%%%%%%%%
\node[left]  at (0, .5*\TOP)  {and \quad $Y_1 = $};
%%%%%%%%%%%%%
\pgftransformxshift{.5cm}
%%%%%%%%%%%%%
\PolesTop[\TOP]
\PolesBottom[0]
\LPole[0,.5*\TOP]
\RPoleO[0,.5*\TOP]
\Over[1,0][-.5,.5*\TOP]
\Under[-.5,.5*\TOP][1,\TOP]
\LPoleO[.5*\TOP,\TOP]
\RPoleO[.5*\TOP,\TOP]
% \foreach \x in {2,...,\K}  {
%	 \draw[thin] (\x,0) -- (\x,\TOP);
%	 }
\Nodes[0][\K]
\Nodes[\TOP][\K]
%%%%%%%%%%%%%
\pgftransformxshift{\K cm}
%%%%%%%%%%%%%
\node[right]  at (0, .5*\TOP)  {.};
\end{tikzpicture}}$$


\centerline{Let $Y_2 = T_1 Y_1 T_1 = \begin{minipage}{2.25cm}{\def\TOP{1}\def\K{2}
\begin{tikzpicture}[line width=1pt , scale=.75]
\PolesTop[\TOP]
\PolesBottom[0]
\LPole[0,.5*\TOP]
\RPoleO[0,.5*\TOP]
\Under[-.5,.5*\TOP][2,\TOP]
\LPoleO[.5*\TOP,\TOP]
\RPoleO[.5*\TOP,\TOP]
	 \draw[over] (1,0) -- (1,\TOP);
\Over[2,0][-.5,.5*\TOP]
\Nodes[0][\K]
\Nodes[\TOP][\K]\end{tikzpicture}}
\end{minipage}$.}}
\end{frame}


\begin{frame}{Exploring our new favorite tensor space, $A \otimes B \otimes C^{\otimes k}$} 
~\\
Products of rectangles:
	$$L((a^c)) \otimes L((b^d)) = \bigoplus_{\lambda \in \Lambda} L(\lambda) \qquad \text{\tiny (multiplicity one!)}$$

where $\Lambda$ is the following set of partitions\only<1-4>{:}\only<5>{\ldots} \\

\vfill

\centerline{
\only<1-4>{\begin{tikzpicture}[xscale=.3, yscale=-.3]
\uncover<2-4>{	\draw (0,0)--(5,0)--(5,4)--(0,4)--(0,0);} 
\uncover<2-3>{	\draw (5,0)--(8,0)--(8,3)--(5,3);
	\node[above] at (2.5,0) {\small $a$}; 	\node[above] at (6.5,0) {\small $b$};
	 \node[left] at (0,2) {\small $c$}; 	\node[right] at (8,1.5) {\small $a$}; }
\uncover<3-4>{	 \draw[structure] (8,.5)--(6.5,.5)--(6.5,2.5)--(6,2.5)--(6,3);}
\uncover<4>{
	 \draw[structure] (2,4)--(2,4.5)--(1.5,4.5)--(1.5,6.5)--(0,6.5);
	 \draw(5,0)--(8,0)--(8,.5) (5,3)--(6,3);
	 \draw(0,4)--(0,6.5);
} \end{tikzpicture}}
\only<5->{
\begin{tikzpicture}[xscale=.2, yscale=-.2]
	\draw (0,0)--(5,0)--(5,4)--(0,4)--(0,0);
	\node at (2.5,2) {\small $(a^c)$};
	\ShiftX{7.5cm}\ShiftY{1cm}
	\node[left] at (0,1) {$\otimes$};
	\Part{2,2}
	\node[right] at (3,1){$=$} ;
	\ShiftX{6cm}\ShiftY{-1cm}
		\Part{7,7} 
		\draw[fill=white] (0,0)--(5,0)--(5,4)--(0,4)--(0,0);
		\ShiftX{10cm}	
		\node[left] at (0,2) {$\oplus$};
		\Part{7,6,0,0,1}
		\draw[fill=white] (0,0)--(5,0)--(5,4)--(0,4)--(0,0);	
		 \ShiftX{10cm}	
		\node[left] at (0,2) {$\oplus$};
		\Part{7,0,0,0,2}
		\draw[fill=white] (0,0)--(5,0)--(5,4)--(0,4)--(0,0);	
		 \ShiftY{7cm}\ShiftX{-15cm}
		\node[left] at (0,2) {$\oplus$};
		\Part{6,6,0,0,1,1}
		\draw[fill=white] (0,0)--(5,0)--(5,4)--(0,4)--(0,0);	
		 \ShiftX{10cm}	
		\node[left] at (0,2) {$\oplus$};
		\Part{6,0,0,0,2,1}
		\draw[fill=white] (0,0)--(5,0)--(5,4)--(0,4)--(0,0);	
		 \ShiftX{10cm}	
		\node[left] at (0,2) {$\oplus$};
		\Part{0,0,0,0,2,2}
		\draw[fill=white] (0,0)--(5,0)--(5,4)--(0,4)--(0,0);	
\end{tikzpicture}
}}

\vfill

\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Brattelli diagram
\newcommand{\BoxPart}[2]{
\draw (0,0)--(5,0)--(5,4)--(0,4)--(0,0);
\coordinate (up) at (5,0); \coordinate (down) at (0,4);
\begin{scope}[shift=(up)]\Part{#1}\end{scope}
\begin{scope}[shift=(down)]\Part{#2}\end{scope}
}
\newcommand{\BoxPART}[2]{
\begin{tikzpicture}[xscale=\PartUNIT, yscale=-\PartUNIT]
	\node at (8,0) {}; \node at (0,7){};
	\BoxPart{#1}{#2}
\end{tikzpicture}
}
\newcommand{\BoxPARTs}[2]{
\begin{tikzpicture}[xscale=\PartUNIT, yscale=-\PartUNIT]
	\node at (8,0) {}; \node at (0,6){};
	\BoxPart{#1}{#2}
\end{tikzpicture}
}
\tikzstyle{EdgeLabel}=[blue!70!black, inner sep=2pt, fill=white]

 \def\PartUNIT{.125}
%Self-contained tikz images for \Part above. 
\newcommand{\PART}[1]{
\begin{matrix}
\begin{tikzpicture}[xscale=\PartUNIT, yscale=-\PartUNIT] 
	\Part{#1}
\end{tikzpicture}
\end{matrix}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%
\begin{frame}{Exploring our new favorite tensor space, $A \otimes B \otimes C^{\otimes k}$} 
$$\!\!\!\!\!\!\!\!\!\!\!\!
\begin{tikzpicture}[xscale=14*\PartUNIT, yscale=-10*\PartUNIT]
\node[right] at (6.5,-.5) {\small $k= 0$};
\uncover<2->{
	\node[right] at (6.5,1) {\small $k=1$};}
\uncover<4->{
	\draw[|-|] (6.5,2.5)--(6.5,4.5);
	\node[right] at (6.5,3.5) {\small $k=2$};}
\coordinate (01) at (3.5,-.5);
\foreach \x in {1, ..., 6}{ \coordinate (1\x) at (\x,1);}
\foreach \x in {1, ..., 6}{ \coordinate (2\x1) at (\x,2.5);}
\foreach \x in {1, ..., 6}{ \coordinate (2\x2) at (\x,3.5);}
\foreach \x in {1, ..., 6}{ \coordinate (2\x3) at (\x,4.5);}
\uncover<2->{
	\draw (01) to [bend left=25] (11);
	\draw (01) to [bend left=10] (12);
	\draw (01) to [bend left=0] (13);
	\draw (01) to [bend right=0] (14);
	\draw (01) to [bend right=10] (15);
	\draw (01) to [bend right=25] (16);
	}
\uncover<3->{	
	\draw 
		(11)--(211) %node[midway,EdgeLabel] {\tiny -$c$}
		(12)--(211) %node[pos=.6, EdgeLabel] {\tiny $a$}
	;
	}
\uncover<4->{	
\draw
	(12)--(221) %node[midway, EdgeLabel] {\tiny -$c$+1}
	(12)--(231) %node[pos=.35, EdgeLabel] {\tiny -$c$-1}
	(13)--(221) %node[pos=.6, EdgeLabel] {\tiny $a$-1}
	(13)--(241) %node[pos=.35, EdgeLabel] {\tiny -$c$-1}
	(14)--(231) %node[pos=.6, EdgeLabel] {\tiny $a$+1}
	(14)--(251) %node[pos=.35, EdgeLabel] {\tiny -$c$+1}
	(15)--(241) %node[pos=.6, EdgeLabel] {\tiny $a$+1}
	(15)--(251) %node[midway, EdgeLabel] {\tiny $a$-1}
	(15)--(261) %node[pos=.4, EdgeLabel] {\tiny -$c$}
	(16)--(261) %node[midway, EdgeLabel] {\tiny $a$}
	;}
\begin{scope}[every node/.style={fill=white, inner sep=1pt}]
	\node at (01) {\begin{tikzpicture}[xscale=\PartUNIT, yscale=-\PartUNIT]
		\draw (0,0)--(5,0)--(5,4)--(0,4)--(0,0);
		\node[right] at (0,2) {\tiny $c$};
		\node[above] at (2.5,0) {\tiny $a$};
		\end{tikzpicture}
};
%
\uncover<3->{
	\node at (211) {\BoxPART{2,2}{1}};
	}
\uncover<4->{	
	\node at (221) {\BoxPART{2,1}{2}};
	\node at (231) {\BoxPART{2,1}{1,1}};
	\node at (241) {\BoxPART{1,1}{2,1}};
	\node at (251) {\BoxPART{1,1}{2,1}};
	\node at (261) {\BoxPART{1}{2,2}};
\draw (11) .. controls (-.55+1,2) and (-.55+1,2.5) .. (212);% node[pos=.8,EdgeLabel]{\tiny $a$+2};
\draw (12) .. controls (-.55+2,2) and (-.55+2,2.5) .. (222);% node[pos=.8,EdgeLabel]{\tiny $a$+2};
\draw (13) .. controls (-.55+3,2) and (-.55+3,2.5) .. (232);% node[pos=.8,EdgeLabel]{\tiny $a$+2};
\draw (14) .. controls (-.55+4,2) and (-.55+4,2.5) .. (242);% node[pos=.8,EdgeLabel]{\tiny $a$-2};
\draw (15) .. controls (-.55+5,2) and (-.55+5,2.5) .. (252);% node[pos=.8,EdgeLabel]{\tiny -$c$+2};
\draw (16) .. controls (-.55+6,2) and (-.55+6,2.5) .. (262);% node[pos=.8,EdgeLabel]{\tiny -$c$+2};
%
	\node at (212) {\BoxPART{3,2}{}};
	\node at (222) {\BoxPART{3,1}{1}};
	\node at (232) {\BoxPART{3}{2}};
	\node at (242) {\BoxPART{1,1,1}{1,1}};
	\node at (252) {\BoxPART{1}{3,1}};
	\node at (262) {\BoxPART{}{3,2}};
\draw (11) .. controls (-.8+1,2) and (-.8+1,3.5) .. (213);% node[pos=.8,EdgeLabel]{\tiny $a$-2};
\draw (12) .. controls (-.8+2,2) and (-.8+2,3.5) .. (223);% node[pos=.8,EdgeLabel]{\tiny $a$-2};
\draw (13) .. controls (-.8+3,2) and (-.8+3,3.5) .. (233);% node[pos=.8,EdgeLabel]{\tiny -$c$+2};
\draw (14) .. controls (-.8+4,2) and (-.8+4,3.5) .. (243);% node[pos=.8,EdgeLabel]{\tiny -$c$-2};
\draw (15) .. controls (-.8+5,2) and (-.8+5,3.5) .. (253);% node[pos=.8,EdgeLabel]{\tiny -$c$-2};
\draw (16) .. controls (-.8+6,2) and (-.8+6,3.5) .. (263);% node[pos=.8,EdgeLabel]{\tiny -$c$-2};
%
	\node at (213) {\BoxPART{2,2,1}{}};
	\node at (223) {\BoxPART{2,1,1}{1}};
	\node at (233) {\BoxPART{2}{3}};
	\node at (243) {\BoxPART{1,1}{1,1,1}};
	\node at (253) {\BoxPART{1}{2,1,1}};
	\node at (263) {\BoxPART{}{2,2,1}};
	}
%
\uncover<2->{
	\node at (11) {\BoxPARTs{2,2}{}};
	\node at (12) {\BoxPARTs{2,1}{1}};
	\node at (13) {\BoxPARTs{2}{2}};
	\node at (14) {\BoxPARTs{1,1}{1,1}};
	\node at (15) {\BoxPARTs{1}{2,1}};
	\node at (16) {\BoxPARTs{}{2,2}};
	}
\end{scope}
\end{tikzpicture}
$$\end{frame}
%%%%%%%%%%%%%



%
%
%
%\begin{frame}{Exploring $L((a^c)) \otimes L((b^d)) \otimes \big(L(\tikz[scale=.2]{\BOX{0}{0}{}})\big)^{\otimes k}$} 
%~\\
%Products of rectangles:
%	$$L((a^c)) \otimes L((b^d)) = \bigoplus_{\lambda \in \Lambda} L(\lambda) \qquad \text{\tiny (multiplicity one!)}$$
%
%where $\Lambda$ is the following set of partitions\only<1-4>{:}\only<5>{\ldots} \\
%
%\vfill
%
%\centerline{
%\only<1-4>{\begin{tikzpicture}[xscale=.3, yscale=-.3]
%\uncover<2-4>{	\draw (0,0)--(5,0)--(5,4)--(0,4)--(0,0);} 
%\uncover<2-3>{	\draw (5,0)--(8,0)--(8,3)--(5,3);
%	\node[above] at (2.5,0) {\small $a$}; 	\node[above] at (6.5,0) {\small $b$};
%	 \node[left] at (0,2) {\small $c$}; 	\node[right] at (8,1.5) {\small $a$}; }
%\uncover<3-4>{	 \draw[structure] (8,.5)--(6.5,.5)--(6.5,2.5)--(6,2.5)--(6,3);}
%\uncover<4>{
%	 \draw[structure] (2,4)--(2,4.5)--(1.5,4.5)--(1.5,6.5)--(0,6.5);
%	 \draw(5,0)--(8,0)--(8,.5) (5,3)--(6,3);
%	 \draw(0,4)--(0,6.5);
%} \end{tikzpicture}}
%\only<5->{
%\begin{tikzpicture}[xscale=.2, yscale=-.2]
%	\draw (0,0)--(5,0)--(5,4)--(0,4)--(0,0);
%	\node at (2.5,2) {\small $(a^c)$};
%	\ShiftX{7.5cm}\ShiftY{1cm}
%	\node[left] at (0,1) {$\otimes$};
%	\Part{2,2}
%	\node[right] at (3,1){$=$} ;
%	\ShiftX{6cm}\ShiftY{-1cm}
%		\Part{7,7} 
%		\draw[fill=white] (0,0)--(5,0)--(5,4)--(0,4)--(0,0);
%		\ShiftX{10cm}	
%		\node[left] at (0,2) {$\oplus$};
%		\Part{7,6,0,0,1}
%		\draw[fill=white] (0,0)--(5,0)--(5,4)--(0,4)--(0,0);	
%		 \ShiftX{10cm}	
%		\node[left] at (0,2) {$\oplus$};
%		\Part{7,0,0,0,2}
%		\draw[fill=white] (0,0)--(5,0)--(5,4)--(0,4)--(0,0);	
%		 \ShiftY{7cm}\ShiftX{-15cm}
%		\node[left] at (0,2) {$\oplus$};
%		\Part{6,6,0,0,1,1}
%		\draw[fill=white] (0,0)--(5,0)--(5,4)--(0,4)--(0,0);	
%		 \ShiftX{10cm}	
%		\node[left] at (0,2) {$\oplus$};
%		\Part{6,0,0,0,2,1}
%		\draw[fill=white] (0,0)--(5,0)--(5,4)--(0,4)--(0,0);	
%		 \ShiftX{10cm}	
%		\node[left] at (0,2) {$\oplus$};
%		\Part{0,0,0,0,2,2}
%		\draw[fill=white] (0,0)--(5,0)--(5,4)--(0,4)--(0,0);	
%\end{tikzpicture}
%}}
%
%\vfill
%
%\end{frame}
%
%


%%%%%%%
\begin{frame}

\def\LevZero{1}
\def\LevOne{2}
\def\LevTwo{3}
\def\LevThree{4}
\def\LevFour{5}
\def\LevFive{6}
\def\TabTwo{7}
\def\TabThree{8}
\def\HRep{9}
\def\HRepB{10}
\def\Cont{11}
\def\ContSa{12}
\def\ContS{13}


$
L\left(\begin{matrix}
	\begin{tikzpicture}[scale=.25]
		\foreach \x in {0,...,4}{\draw (\x,0)--(\x,-3);}
		\foreach \y in {0,...,3}{\draw (0,-\y)--(4,-\y);}
	\end{tikzpicture}\end{matrix}\right)
\otimes 
	L\left(\begin{matrix}
		\begin{tikzpicture}[scale=.25]
		\foreach \x in {0,...,2}{\draw (\x,0)--(\x,-2);}
		\foreach \y in {0,...,2}{\draw (0,-\y)--(2,-\y);}
	\end{tikzpicture}
\end{matrix}\right)
\uncover<\LevOne->{
\otimes 
	L\left(\begin{matrix}
		\begin{tikzpicture}[scale=.25]
		\Part{1}
	\end{tikzpicture}
\end{matrix}\right)
}
\uncover<\LevTwo->{
\otimes 
	L\left(\begin{matrix}
		\begin{tikzpicture}[scale=.25]
		\Part{1}
	\end{tikzpicture}
\end{matrix}\right)
}
\uncover<\LevThree->{
\otimes 
	L\left(\begin{matrix}
		\begin{tikzpicture}[scale=.25]
		\Part{1}
	\end{tikzpicture}
\end{matrix}\right)
}
\uncover<\LevFour->{
\otimes 
	L\left(\begin{matrix}
		\begin{tikzpicture}[scale=.25]
		\Part{1}
	\end{tikzpicture}
\end{matrix}\right)
}
\uncover<\LevFive->{
\otimes 
	L\left(\begin{matrix}
		\begin{tikzpicture}[scale=.25]
		\Part{1}
	\end{tikzpicture}
\end{matrix}\right)
}
$

\small
\centerline{$
\uncover<\ContS->{
	\begin{matrix}
		~&~&~\\
		Y_1 \!\!\!&\mapsto&\!\!\! t^{\alert{5.5}}\\
		Y_2 \!\!\!&\mapsto&\!\!\! t^{\alert{3.5}}\\
		Y_3 \!\!\!&\mapsto&\!\!\! t^{\alert{-4.5}}\\
		Y_4 \!\!\!&\mapsto&\!\!\! t^{\alert{-5.5}}\\
		Y_5 \!\!\!&\mapsto&\!\!\! t^{\alert{-2.5}}
	\end{matrix}}
\qquad 
\begin{matrix}
\begin{tikzpicture}[xscale=.4, yscale=-.4]
	\uncover<\Cont>{
		\foreach \x in {0,1,...,6} {\draw[structure] (\x, 0) to (\x-.5, -.5) node[inner sep=1pt, above left] {\tiny \x};}
		\foreach \x in {1,2,...,5} {\draw[structure] (0, \x) to (-.5, \x-.5) node[inner sep=1pt, above left] {\tiny -\x};}}
	\uncover<\ContSa->{
		\node[above, structure] at (3.5,-1.5) {\scriptsize Shift by $\half (a-c+b-d)$};
		\draw[very thick, ->, structure] (2,-1.5)--(5,-1.5);
		\foreach \x in {0,1,...,6} {\draw[structure] (\x+.5, 0) to (\x, -.5) node[inner sep=1pt, above left] {\tiny \x};}
		\foreach \x in {1,2,...,6} {\draw[structure] (0, \x-.5) to (-.5, \x-1) node[inner sep=1pt, above left] {\tiny -\x};}}
	\filldraw[structure!10] (0,0)--(0,4)--(1,4)--(1,3)--(4,3)--(4,2)--(5,2)--(5,1)--(6,1)--(6,0);
	\Part{6,5,4,1}
	\uncover<\LevOne->{\BOX{6}{0}{1}}
	\uncover<\LevTwo->{\BOX{5}{1}{2}}
	\uncover<\LevThree->{\BOX{0}{4}{3}}
	\uncover<\LevFour->{\BOX{0}{5}{4}}
	\uncover<\LevFive->{\BOX{1}{3}{5}}
	\draw[structure, line width=1pt] (0,0)--(4,0)--(4,3)--(0,3)--(0,0);
	\draw[structure!50, line width=1pt,dotted] 
		(4,0)--(6,0)--(6,2)--(4,2)--(4,0)
		(0,3)--(0,5)--(2,5)--(2,3)--(0,3);
\ShiftX{8.5cm}
	\uncover<\TabTwo->{
		\filldraw[structure!10] (0,0)--(0,4)--(1,4)--(1,3)--(4,3)--(4,2)--(5,2)--(5,1)--(6,1)--(6,0);
		\Part{6,5,4,1}
		\BOX{6}{0}{5}	\BOX{5}{1}{4}	\BOX{0}{4}{1}
		\BOX{0}{5}{3}	\BOX{1}{3}{2}
		\draw[structure, line width=1pt] (0,0)--(4,0)--(4,3)--(0,3)--(0,0);
		\draw[structure!50, line width=1pt,dotted] 
		(4,0)--(6,0)--(6,2)--(4,2)--(4,0)
		(0,3)--(0,5)--(2,5)--(2,3)--(0,3);}
\end{tikzpicture}\end{matrix}
\qquad
\uncover<\ContS->{
	\begin{matrix}
		~&~&~\\
		Y_1 \!\!\!&\mapsto&\!\!\! t^{\alert{5.5}}\\
		Y_2 \!\!\!&\mapsto&\!\!\! t^{\alert{3.5}}\\
		Y_3 \!\!\!&\mapsto&\!\!\! t^{\alert{-4.5}}\\
		Y_4 \!\!\!&\mapsto&\!\!\! t^{\alert{-5.5}}\\
		Y_5 \!\!\!&\mapsto&\!\!\! t^{\alert{-2.5}}
	\end{matrix}}
$}
\centerline{$
\uncover<\ContS->{
	\begin{matrix}
		Y_1 \!\!\!&\mapsto&\!\!\! t^{\alert{-5.5}}\\
		Y_2 \!\!\!&\mapsto&\!\!\! t^{\alert{2.5}}\\
		Y_3 \!\!\!&\mapsto&\!\!\! t^{\alert{4.5}}\\
		Y_4 \!\!\!&\mapsto&\!\!\! t^{\alert{3.5}}\\
		Y_5 \!\!\!&\mapsto&\!\!\! t^{\alert{5.5}}
	\end{matrix}}
\qquad 
\begin{matrix}
\begin{tikzpicture}[xscale=.4, yscale=-.4]
	\uncover<\TabThree->{
		\foreach \x in {5} {\draw[white] (0, \x) to (-.5, \x-.5) node[inner sep=1pt, above left,white] {\tiny -\x};}
		\filldraw[structure!10] (0,0)--(0,5)--(1,5)--(1,4)--(2,4)--(2,3)--(4,3)--(4,1)--(5,1)--(5,0);
		\Part{5,4,4,2,1}
		\BOX{6}{0}{5}	\BOX{5}{1}{4}	\BOX{0}{5}{1}
		\BOX{5}{0}{3}	\BOX{4}{1}{2}
		\draw[structure, line width=1pt] (0,0)--(4,0)--(4,3)--(0,3)--(0,0);
		\draw[structure!50, line width=1pt,dotted] 
			(4,0)--(6,0)--(6,2)--(4,2)--(4,0)
			(0,3)--(0,5)--(2,5)--(2,3)--(0,3);
	\ShiftX{8.5cm}
		\filldraw[structure!10] (0,0)--(0,5)--(1,5)--(1,3)--(4,3)--(4,2)--(5,2)--(5,0);
		\Part{6,5,4,1,1}
		\BOX{6}{0}{5}	\BOX{5}{1}{4}	\BOX{0}{5}{1}
		\BOX{5}{0}{3}	\BOX{1}{3}{2}
		\draw[structure, line width=1pt] (0,0)--(4,0)--(4,3)--(0,3)--(0,0);
		\draw[structure!50, line width=1pt,dotted] 
		(4,0)--(6,0)--(6,2)--(4,2)--(4,0)
		(0,3)--(0,5)--(2,5)--(2,3)--(0,3);}
\end{tikzpicture}\end{matrix}
\qquad
\uncover<\ContS->{
	\begin{matrix}
		Y_1 \!\!\!&\mapsto&\!\!\! t^{\alert{5.5}}\\
		Y_2 \!\!\!&\mapsto&\!\!\! t^{\alert{3.5}}\\
		Y_3 \!\!\!&\mapsto&\!\!\! t^{\alert{-4.5}}\\
		Y_4 \!\!\!&\mapsto&\!\!\! t^{\alert{-5.5}}\\
		Y_5 \!\!\!&\mapsto&\!\!\! t^{\alert{-2.5}}
\end{matrix}}
$}


\uncover<\LevFive->{\alert{($*$)} $H_k$ representations in tensor space are labeled by certain partitions $\lambda$.\\}
\uncover<\HRep->{\alert{($*$)} Basis labeled by tableaux 
from {\sl some} partition $\mu$ in $(a^c) \otimes (b^d)$ to $\lambda$.}

\uncover<\HRepB->{\alert{($*$)} Calibrated}\uncover<\ContS->{: $Y_i$ acts by $t$ to the shifted content of $\mathrm{box}_i$.}
\end{frame}

\begin{frame}{From $\{$partitions in tensor space$\}$ to $\{$box arrangements$\}$}

$$\begin{tikzpicture}[xscale=.5, yscale=-.5]
\uncover<6->{
	\filldraw[black!5] (0,3)--(0,5)--(1,5)--(1,4)--(2,4)--(2,3);
	\filldraw[black!5] (4,2)--(6,2)--(6,0)--(5,0)--(5,1)--(4,1);
	}
\uncover<3->{%Blacked out boxes
	\filldraw[black!30] (0,0)--(0,3)--(4,3)--(4,1)--(5,1)--(5,0);
	}
\uncover<4->{
		\foreach \x in {0,1,2,3} {\draw[structure] (\x+.5, 0) to (\x, -.5) node[inner sep=1pt, above left] {\tiny \x};}
		\foreach \x in {1,2,3} {\draw[structure] (0, \x-.5) to (-.5, \x-1) node[inner sep=1pt, above left] {\tiny -\x};}
		\node[structure] at (4,-.5) {$\dots$};\node[structure] at (-.5,3) {$\vdots$};
		\draw[red] (4,2)--(4.5,2.5) node[inner sep=1pt, below right] {\tiny $r_1$};
		\draw[red] (2,3)--(2.5,3.5) node[inner sep=1pt, below right] {\tiny -$r_1$};
		}
\uncover<4-6>{
		\draw[red] (0,5)--(-.5,4.5) node[inner sep=1pt, above left] {\tiny -$r_2$};
		\draw[red] (6,0)--(5.5,-.5) node[inner sep=1pt, above left] {\tiny $r_2$};
}
\uncover<7->{
		\draw[red] (-1,4)--(-1.5,3.5) node[inner sep=1pt, above left] {\tiny -$r_2$};
		\draw[red] (5,-1)--(4.5,-1.5) node[inner sep=1pt, above left] {\tiny $r_2$};
}
\uncover<1->{\Part{7,6,4,2,1,1}}%Main partition
\uncover<7->{
	\draw (5,0)--(5,-1)--(6,-1)--(6,0);
	\draw (0,5)--(-1,5)--(-1,4)--(0,4);
	}
\uncover<2->{
	\draw[structure, line width=1pt] (0,0)--(4,0)--(4,3)--(0,3)--(0,0);
	\draw[structure, line width=1pt,dotted] 
		(4,0)--(6,0)--(6,2)--(4,2)--(4,0)
		(0,3)--(0,5)--(2,5)--(2,3)--(0,3);
	}
\begin{scope}[every node/.style={structure}]
	\node at (5.5,.5) {1};
	\node at (5.5,1.5) {2};
	\node at (6.5,.5) {3};
	\node at (1.5,3.5) {4};
	\node at (.5,5.5) {5};
	\end{scope}
\uncover<8->{
	\node at (.5,4.5) {-1};
	\node at (.5,3.5) {-2};
	\node at (-.5,4.5) {-3};
	\node at (4.5,1.5) {-4};
	\node at (5.5,-.5) {-5};
}
\uncover<5->{%Markers
	\filldraw[structure, fill=red]  (6,0) circle (5pt);
	\filldraw[structure, fill=red]  (4,2) circle (5pt);
	\filldraw[structure, fill=red]  (2,3) circle (5pt);
	\filldraw[structure, fill=red]  (0,5) circle (5pt);
	}
\ShiftX{12cm}\ShiftY{1cm}
\uncover<9->{
	\draw[->, thick] (-4.5,2)--(-2.5,2);
	\begin{scope}[structure, every node/.style={above left, inner sep=1pt}]
		\draw[red] (0,0)--(-.5,-.5) node {\tiny $r_2$};
		\draw (0,1)--(-.5,.5) node {\tiny $4.5$};
		\draw (0,2)--(-.5,1.5) node {\tiny $3.5$};
		\draw (-1,2)--(-1.5,1.5) node {\tiny $2.5$};
	\end{scope}
	\draw (0,0)--(0,3)--(1,3)--(1,0)--(0,0) (0,1)--(2,1)--(2,2)--(-1,2)--(-1,3)--(0,3);
	\node at (.5,.5) {-5};
	\node at (.5,1.5) {1};
	\node at (.5,2.5) {2};
	\node at (1.5,1.5) {3};
	\node at (-.5,2.5) {-4};
		\filldraw[structure, fill=red]  (1,1) circle (5pt);
}
\end{tikzpicture}
$$
\uncover<3->{$\begin{tikzpicture}[scale=.25] \filldraw[black!30] (0,0)--(1,0)--(1,1)--(0,1); \draw (0,0)--(1,0)--(1,1)--(0,1)--(0,0);\end{tikzpicture} = $ \small boxes that must appear in the partition at level 0.}

\uncover<4->{$$
\gamma(Y_1) = t^{\color{structure}4.5}, ~ \gamma(Y_2) = t^{\color{structure}3.5}, ~ \gamma(Y_3) = t^{\color{structure}r_2}, ~ \gamma(Y_4) = t^{\color{structure}-2.5}, ~ \gamma(Y_5) = t^{\color{structure}-r_2}.
$$}\uncover<9->{
\centerline{versus}
$$
\gamma(Y_1) = t^{\color{structure}4.5}, ~ \gamma(Y_2) = t^{\color{structure}3.5}, ~ \gamma(Y_3) = t^{\color{structure}r_2}, ~ \gamma(Y_4^{-1}) = t^{\color{structure}2.5}, ~ \gamma(Y_5^{-1}) = t^{\color{structure}r_2}.
$$
}

\end{frame}

{\usebackgroundtemplate{\includegraphics[width=\paperwidth]{blackboard}}
\begin{frame}[t]
\color{white} \centerline{\Large Thanks!}
\end{frame}}


\end{document}